157 Identify the domain of a logarithmic function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as [latex]y={b}^{x}\\[/latex] for any real number x and constant [latex]b>0\\[/latex], [latex]b\ne 1\\[/latex], where

• The domain of y is [latex]\left(-\infty ,\infty \right)\\[/latex].
• The range of y is [latex]\left(0,\infty \right)\\[/latex].

In the last section we learned that the logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] is the inverse of the exponential function [latex]y={b}^{x}\\[/latex]. So, as inverse functions:

• The domain of [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] is the range of [latex]y={b}^{x}\\[/latex]:[latex]\left(0,\infty \right)\\[/latex].
• The range of [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] is the domain of [latex]y={b}^{x}\\[/latex]: [latex]\left(-\infty ,\infty \right)\\[/latex].

Transformations of the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range of [latex]y={b}^{x}\\[/latex]. Similarly, applying transformations to the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, consider [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)\\[/latex]. This function is defined for any values of x such that the argument, in this case [latex]2x - 3\\[/latex], is greater than zero. To find the domain, we set up an inequality and solve for x:

In interval notation, the domain of [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)\\[/latex] is [latex]\left(1.5,\infty \right)\\[/latex].